1 convergence of experimental and modeling studies1.2 review of various model systems 3treatment...

1

Convergence of Experimental and Modeling Studies Vikas Mittal

1.1 Introduction

Experimental results on composite properties are generally modeled using differ-ent fi nite element and micromechanical models to gain further insights into the experimental fi ndings. Such models are also useful in predicting the properties of same or similar materials, thus eliminating the need for synthesizing each and every composite fi rst to ascertain its properties. A number of precautions are, however, necessary to avoid discrepancies in the model outcome, for example, the model used should not have unrealistic assumptions, and the experimental results should be in plenty to have an accurate model. The following sections present some examples of modeling and prediction of polymer clay nanocomposite proper-ties using micromechanical, fi nite element, and factorial design methods.

1.2 Review of Various Model Systems

A number of micromechanical models have been developed over the years to predict the mechanical behavior of particulate composites [1 – 4] . The Halpin – Tsai model has received special attention owing to better prediction of the properties for a variety of reinforcement geometries. The relative tensile modulus is expressed as

E E/ 1 / 1m f f= + −( ) ( )ζηϕ ηϕ

where E and E m correspond to the elastic moduli of composite and matrix, respec-tively, ζ represents the shape factor, which is dependent on fi ller geometry and loading direction and φ f is the inorganic volume fraction. η is given by the expression

η ζ= − +( ) ( )E E E Ef m f m/ 1 / /

where E f is the modulus of the fi ller. The η values need to be correctly defi ned in order to have better prediction of the properties. For the oriented discontinuous

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Modeling and Prediction of Polymer Nanocomposite Properties, First Edition. Edited by Vikas Mittal.© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Convergence of Experimental and Modeling Studies

ribbon or lamellae, it is estimated to be twice the aspect ratio. It has been reported to overpredict the stiffness in this case; therefore, its value was reported be 2/3 times the aspect ratio [5] . Nevertheless, several assumptions prevent the theory to correctly predict the stiffness of the layered silicate nanocomposites. Assumptions like fi rm bonding of fi ller and matrix, perfect alignment of the platelets in the matrix, and uniform shape and size of the fi ller particles in the matrix make it very diffi cult to correctly predict the nanocomposites properties. Incomplete exfoliation of the nanocomposites, thus, the presence of a distribution of tactoid thicknesses, is another concern. The model has recently been modifi ed to accom-modate the effect of incomplete exfoliation and misorientation of the fi ller, but the effect of imperfect adhesion at the surface still needs to be incorporated [6, 7] .

As a case study, tensile properties of polypropylene ( PP ) nanocomposites con-taining dioctadecyldimethylammonium - modifi ed montmorillonite (2C18 • M880) using different fi ller inorganic volume fraction were modeled using these micro-mechanical approaches [8] . The modulus of the composites linearly increased with volume fraction with an increase of 45% at 4 vol% as compared to the pure PP. As shown in Figure 1.1 a, the data were fi tted to the conventional Halpin – Tsai equation with η = 1, which gives a value of 10.1 for ζ , indicating that possibly in these nanocomposites it cannot be simply taken as twice the aspect ratio as gener-ally used [9] . To account for the incomplete fi ller exfoliation and the presence of tactoid stacks in the composites, thickness of the particle was explained by the following equation:

t d n tparticle 1 platelet1= − +00 ( )

where d 001 is the basal plane spacing of 001 plane, n is the number of the platelets in the stack, and t platelet is the thickness of one platelet in the pristine montmorillo-nite. Thus, in this approach, fi ller particles were replaced by the stacks of fi ller platelets [6] . Applying this treatment to the Halpin – Tsai equation, different curves have been generated based on the number of platelets present in the stack as shown in Figure 1.1 b. As is evident from the fi gure, the experimental value of relative tensile modulus for 1 vol% OMMT composites lies near to theoretical curve with 50 platelets in the stack, but the composites with higher volume fractions of the fi ller could not follow the predicted rise in the modulus. The observed behavior underlines another important limitation of the theoretical models for their inability to take into account the possible decrease in d - spacing with increasing volume frac-tion. Besides, the effect of misoriented platelets on the modulus also needs to be incorporated in the model. As can be seen in the SEM micrographs in Figure 1.2 , for the 3 vol% 2C18 • M880 OMMT - PP nanocomposites, the fi ller platelets can be safely treated as random and misaligned in the matrix. Figure 1.3 a shows the result-ing comparison when the effects of incomplete exfoliation combined with the platelets misalignment considerations were incorporated in the Halpin – Tsai model for random 3D platelets [5] . As can be seen the number of platelets in the stack for 1 vol% composites now lie between 30 and 50 ( ∼ 40). Brune and Bicerano have also refi ned the predictions for the behavior of nanocomposites based on the combina-tion of incomplete exfoliation and misorientation [7] . Comparing the suggested

1.2 Review of Various Model Systems 3

treatment with the experimental data, Figure 1.3 b showed that the number of plate-lets in the stacks in 1 vol% composite was observed to be between 20 and 25, which gives an aspect ratio of about 15 for these composites. However, one major limita-tion of the mechanical models is the assumption of perfect adhesion at the inter-face, whereas the polyolefi n composites studied in fact lack this adhesion, as only weak van der Waals forces can exist in the studied polymer organic monolayer systems. The theoretical results predicted above were therefore only able to match the experimental results of polar polymers due to the same reason [10, 11] .

Figure 1.1 (a) Relative tensile modulus of PP nanocomposites plotted as a function of inorganic volume fraction. The solid line represents the fi tting using the unmodifi ed Halpin – Tsai equation. (b) Relative tensile

modulus of the above - mentioned composites ( � : - experimental) compared with the values considering different number of platelets in the stack. Reproduced from reference 8 with permission (Sage Publishers).

(a)

(b)

RR

I

I


Figure 1.2 SEM micrographs of 3 vol% PP nanocomposites. Reproduced from Ref. [8] with permission (Sage Publishers).

(a)

(b)


Nicolais and Nicodemo [12] suggested a simple model to predict the tensile strength of the fi lled polymers described by the equation

σ σ ϕ/ 11 1 P2= − P

where P 1 is stress concentration - related constant with a value of 1.21 for the spheri-cal particles having no adhesion with the matrix and P 2 is geometry - related con-stant with a value of 0.67 when the sample fails by random failure. The yield strength, yield strain, and stress at break for the 2C18 • M880 OMMT - PP com-posites as a function of inorganic fi ller volume fraction have been plotted in Figure 1.4 . The yield strength decayed with augmenting the fi ller volume fraction,

Figure 1.3 Relative tensile modulus of PP nanocomposites at different inorganic volume fraction ( � : - experimental) compared with the values considering different number

of platelets in the stack applying the platelet misorientation corrections. Reproduced from Ref. [8] with permission (Sage Publishers).

RR

I

I

(a)

(b)


indicating the lack of adhesion at the interface and brittleness as shown in Figure 1.4 a. As described earlier, with the addition of low - molecular - weight compatibiliz-ers, an increase in the yield strengths were reported probably due to better adhe-sion, higher extents of delamination, and plasticization effects. The platelets in the present case, which may have been only kinetically trapped, also lead to straining of the confi ned polymer chains. Fitting the values of yield strength in the Nicolais and Nicodemo model yielded P 1 as 2.30 and P 2 as 0.63, thus deviating from the values marked for the spherical particles [9] . The stress at break also decreased nonlinearly with fi ller volume fraction owing to similar reasons and the presence of tactoids. The fi tting of stress at break values (Figure 1.4 b) in the model yielded P 1 and P 2 as 6.13 and 1.03, respectively, which shows higher deviation from the spherical particle predictions. Nielsen [13] suggested that the strain can be pre-dicted by the simple equation as

ε ε ϕc m f1 3/ 1= − /

Figure 1.4 (a) Relative yield strength, (b) relative stress at break, and (c) relative yield strain of PP nanocomposites plotted as a function of inorganic volume fraction. The

solid lines represent the fi tting using the theoretical equations, whereas the dotted line serves simply as a guide. Reproduced from Ref. [8] with permission (Sage Publishers).

(b) (a)

Rel

ativ

e yi

eld

stre

ngth

Rel

ativ

e yi

eld

stra

in

Rel

ativ

e st

ress

at b

reak

Inorganic volume fraction



(c)


where ε c and ε m are the yield strains of the composite and matrix, respectively, and φ f is the fi ller volume fraction. It was assumed that the polymer breaks at the same elongation in the fi lled composite as the bulk unfi lled polymer does. The much lower experimental values (Figure 1.4 c) agree with the lack of adhesion as sug-gested above and the strain hardening of the confi ned polymer. It also indicates that the brittleness increased on increasing the fi ller volume fraction.

Figure 1.5 a shows a typical fi nite element model of round platelets with an aspect ratio of 50 at 3 vol% loading, while Figure 1.5 b presents a 2D - cut through the center of the model [14] . The solid lines in Figure 1.6 a represent the numerical predictions for the relative permeability of composites as a function of increasing volume fraction of misaligned platelets with an aspect ratio of 50 or 100. As noted, there is excellent agreement between the experimentally measured oxygen perme-ability and the numerical predictions up to ca. 3 vol%. Above this concentration, it seems that the number of exfoliated layers decreases, leading to a lower average aspect ratio in both epoxy ( EP ) and polyurethane ( PU ) composites. An average

Figure 1.5 (a) A computer model compris-ing 50 randomly distributed and oriented round platelets with an aspect ratio of 50 at 3 vol% loading, periodic boundary conditions

applied; (b) cross section through the center of the model. Reproduced from Ref. [14] with permission (Wiley).

(a)

(b)


aspect ratio of the montmorillonite platelets in nanocomposites can be estimated from the relative permeability at 3 vol% loading. The effect of misalignment on the barrier performance of platelets with different aspect ratios at 3 vol% loading, as predicted by computer models, is shown in Figure 1.6 b.With increasing aspect ratio, it becomes necessary to align the platelets in order not to lose their effective-ness. Similarly, Figure 1.7 plots the comparison of measured permeability through polypropylene nanocomposites with numerical predictions for composites of par-allel oriented and misaligned disk - shaped impermeable inclusions with aspect ratios (diameter/thickness) 30 and 100, respectively [15] . From this comparison, a macroscopic average of the aspect ratio for the inclusions is estimated to be between 30 and 100. However, a more precise estimation can only be made when the degree of orientation is experimentally determined and an orientation - dependent term is included in the numerical calculation.

Figure 1.6 Dependence of the gas permea-tion through nanocomposites on the inorganic volume fraction, aspect ratio, and orientation of the platelets: (a) comparison between the measured relative oxygen permeability in EP - and PU - nanocomposites

and numerical predictions; (b) infl uence of misalignment on the performance of platelets as permeation barrier at 3 vol% loading as predicted numerically. Reproduced from Ref. [14] with permission (Wiley).

(a)

(b)

Inorganic volume fraction (f)

Aspect ratio (a)

misaligned

aligned

0 50 100 150 200

0.00

1.0

0.8

0.6

0.4

0.2

0.0

0.02 0.04 0.06

EPPU

a = 50

Rel

ativ

e tr

ansm

issi

on r

ate

(TC/T

P)

Rel

ativ

e tr

ansm

issi

on r

ate

(TC/T

P)

a = 100

1.0

0.8

0.6


Figures 1.8 and 1.9 also demonstrate the possibility of modeling and prediction of polyethylene clay nanocomposite properties using mixture design methods. Examples of both oxygen permeation as well as tensile modulus as a function of different amounts of different components polymer, organically modifi ed montmo-rillonite and compatibilizer have been shown. Like conventional models, which depend on oversimplifi ed assumptions, these models do not suffer from these

Figure 1.7 Relative permeability of the 2C18 – M880 - PP nanocomposites as a function of the inorganic volume fraction. The lines represent numerical predictions for composites of parallel oriented and

misaligned disk - shaped impermeable inclusions with aspect ratio (diameter/thickness) of 30 and 100, respectively. Reproduced from Ref. [15] with permission (Wiley).

a = 100 misaligned

Inorganic volume fraction0.00

Rel

ativ

e pe

rmea

bilit

y1.0

0.9

0.8

0.7

0.60.01 0.02 0.03 0.04

a = 30 aligned

Figure 1.8 Mixture plot for the prediction of oxygen permeation of polyethylene nanocompos-ites with different amounts of components: polymer ( P ), organically modifi ed montmorillonite ( OM ) and compatibilizer (Compat).

P

81

100

OM19

0

Compat19

0

30–3535–4040–45

Oxygenpermeation

45–5050–55> 55

< 2525–30

Mixture contour plot of oxygen permeation(component amounts)


Figure 1.9 Mixture plot for the prediction of tensile modulus of polyethylene nanocomposites with different amounts of components: polymer (P), organically modifi ed montmorillonite (OM), and compatibilizer (Compat).

P

81

100

OM19

0

Compat19

0

Tensile

1100–12001200–13001300–1400

modulus

1400–1500> 1500

< 10001000–1100

Mixture contour plot of tensile modulus(component amounts)

limitations and can still predict the composite properties using a set of simple equations.

References

1 Kerner , E.H. ( 1956 ) Proc. Phys. Soc. , B69 , 808 .

2 Hashin , Z. , and Shtrikman , S. ( 1963 ) J. Mech. Phys. Solids , 11 , 127 .

3 Halpin , J.C. ( 1969 ) J. Compos. Mater. , 3 , 732 .

4 Halpin , J.C. ( 1992 ) Primer on Composite Materials Analysis , Technomic , Lancaster .

5 van Es , M. , Xiqiao , F. , van Turnhout , J. , and van der Giessen , E. ( 2001 ) Specialty Polymer Additives: Principles and Application (eds S. Al - Malaika , A.W. Golovoy , and C.A. Wilkie ), Blackwell Science , CA Melden, MA , pp. 391 – 414 .

6 Fornes , T.D. , and Paul , D.R. ( 2003 ) Polymer , 44 , 4993 .

7 Brune , D.A. , and Bicerano , J. ( 2002 ) Polymer , 43 , 369 .

8 Mittal , V. ( 2007 ) J. Thermoplastic Compos. Mater. , 20 , 575 .

9 Osman , M.A. , Rupp , J.E.P. , and Suter , U.W. ( 2005 ) Polymer , 46 , 1653 .

10 Luo , J.J. , and Daniel , I.M. ( 2003 ) Compos. Sci. Technol. , 63 , 1607 .

11 Wu , Y.P. , Jia , Q.X. , Yu , D.S. , and Zhang , L.Q. ( 2004 ) Polym. Test. , 23 , 903 .

12 Nicolais , L. , and Nicodemo , L. ( 1973 ) Polym. Eng. Sci. , 13 , 469 .

13 Nielsen , L.E. ( 1966 ) J. Appl. Polym. Sci. , 10 , 97 .

14 Osman , M.A. , Mittal , V. , and Lusti , H.R. ( 2004 ) Macromol. Rapid Commun. , 25 , 1145 .

15 Osman , M.A. , Mittal , V. , and Suter , U.W. ( 2007 ) Macromol. Chem. Phys. , 208 , 68 .

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